MultiDiFluxMultiDiFlux©

Analysis of Concentration Profiles of Multi-component Diffusion couples

forInterdiffusion Fluxes and Interdiffusion Coefficients

AndDiffusion Paths

M. A. Dayananda, Purdue UniversityE-mail: dayanand@ecn.purdue.edu

QuantumSemiconductorAlgorithms

Dr. L. R. Ram-Mohan, QSAE-mail: lrram@wpi.edu

NIST workshop on Multicomponent Diffusion – April 1-2, 2004

Analysis*

•Concentration profiles that develop in a solid-solid, ternary diffusion couple can be analyzed directly for ternary interdiffusion coefficients defined as average values over selected regions in the diffusion zone.

•These coefficients are calculated from an integration of interdiffusion fluxes which are calculated directly from experimental concentration profiles.

* M. A. Dayananda and Y. H. Sohn, Metall. Mater. Trans. A.,30A (1999) pp. 535-543.

Determination of Interdiffusion Fluxes*

The interdiffusion fluxes of all components can be evaluated directly from the concentration profiles of a solid-solid diffusion couple at any section x in an n-component system without the need for interdiffusioncoefficients.

˜ J i x( ) = 12t

x - xo( )dCiCi ±∞( )

Ci x( )

∫ (i =1,2,...,n)

* M. A. Dayananda and C. W. Kim, Metall. Trans.,10A (1979) 1333.

Interdiffusion Flux

Calculation from Experimental Concentration Profiles[1,2]

Generalized Fick’s Law

1 M.A. Dayananda and C.W. Kim, Metall. Trans. A, 10A 1333-1339 (1979).2 M.A. Dayananda, Metall. Trans. A, 14A 1851-1858 (1983).3 L. Onsager, Phys. Rev., 37 405-426 (1931).4 L. Onsager, Phys. Rev., 38 2265-2279 (1931).

( )1,...,2,1~~ 1

1

−=∂

∂−= ∑

−

=

nix

CDJ j

n

j

niji

[3,4]:

( ) ( )( )

( )nidCxxt

xJ i

xC

CCoi

i

ii

,...,2,121~

or

=−= ∫−+

Flux equations to include molar volume changes

)....2,1(*)*1(* )1(*

2)*( nidxx mV

iYiYdxx

mViY

iYtiC

xMiJ =∫

∞−+∫ ∞−−∆

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

Yi= (Ci - Ci+) / (Ci

- - Ci+)

E+D-A=B+C+D

=(x*- xo)

Integration of Interdiffusion Fluxes

On the basis of Onsager’s formalism,

˜ J i = − ˜ D i13 ∂C1

∂x− ˜ D i2

3 ∂C2

∂x (i =1,2)

˜ J idx = − ˜ D i13dC1

C1 x1( )

C1 x2( )

∫x1

x2

∫ − ˜ D i23 dC2

C2 x1( )

C2 x2( )

∫ (i = 1,2)

= ˜ D i13 C1 x1( )−C1 x2( )[ ]

+ ˜ D i23 C2 x1( )−C2 x2( )[ ] (i = 1,2)

and are the average values of main and cross interdiffusion coefficients.˜ D i13 ˜ D i2

3

Multiplying both sides by (x-xo) and integrating over a selected region, x1 to x2 :

˜ J i x − xo( )dxx1

x2

∫ = − ˜ D i13 x − xo( )dC1

C1 x1( )

C1 x2( )

∫

− ˜ D i23 x − xo( )dC2

C2 x1( )

C2 x2( )

∫ (i = 1,2)

˜ J i x − xo( )dxx1

x2

∫ = 2t ˜ D i13 ˜ J 1 x1( )− ˜ J 1 x2( )[ ]{

+ ˜ D i23 ˜ J 2 x1( )− ˜ J 2 x2( )[ ]} (i = 1,2)

Determination of Coefficients

˜ D ij3 (i, j = 1,2)

0

(-) (+)xo

S3

x2x1distance, x

J i(x

-xo)

~In

terd

iffus

ion

Flu

x, J

i~

0 (-) (+)xo

(Ji)xo~

x2x1

S2

Ci(x)

Cio

xo x (+)(-)

Con

cent

ratio

n, C

i

Ci+

Ci-

S1

S1 = x − xo( )dCiCi

+

Ci x( )

∫ ; S2 = ˜ J idxx1

x2

∫ ; S3 = ˜ J i x − xo( )dxx1

x2

∫

˜ D ij3 = ˜ D ij

3dCjCj x1( )

Cj x2( )

∫ dCjCj x1( )

Cj x2( )

∫ (i = 1,2)

MultiDiflux Program

DATA_INPUT FILENumber of Components: 2 or more

Single Phase or Multiphase: 0 or 1 (switch)

Index of Dependent Concentration: 3 for ternary

Data file: Ci vs. x

Diffusion time: t

Number of Interpolation elements for smoothening the experimental data with Hermite polynomials : Variable

Number of diffusion zones: Number of phase layers

Diffusion regions into which each zone is divided for the calculation of interdiffusion coefficients: Variable

Main Output Files

Expt_.out – Ci vs. x plots

Conc_interp.out – Interpolated profiles

Flux.out - Ji vs. x plots (calculated)

Conc_deriv.out- Derivatives of Ci

Deriv_nodes_data.out: Derivatives of Ci at both ends of each interpolation region

Diff.out - Dij coefficients (calculated)

Matano0.out – Locations of Matano plane

These files are displayed through Gnuplot.

Cu-Isoactivity Couples in the Cu-Ni-Zn System

C.W. Kim and M.A. Dayananda, Metall. Trans. A, 15A, (1984) pp. 649-657

Cu-Isoactivity Cu-Ni-Zn Couple, a10 vs a13

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 100 200 300 400 500 600

Ato

m F

ract

ion

x-coordinate (µm)

Concentration Profiles

Red: Component 1 Green: Component 2 Blue: Component 3

Matano Plane

Zn

Cu

Ni

Matano Plane

ZFP

Area AArea B

Cu-Isoactivity Cu-Ni-Zn Couple, a10 vs a13

ZFP for Cu

Zn

Cu

Ni

Flux Profiles for the Couple, a10 vs a13

-2.5e-05

-2e-05

-1.5e-05

-1e-05

-5e-06

0

5e-06

1e-05

1.5e-05

2e-05

2.5e-05

0 100 200 300 400 500 600

J_i (

atom

frac

.-µm

/sec

)

x-coordinate (µm)

Flux Profiles

Red: Component 1 Green: Component 2 Blue: Component 3

#============================================================# This file shows the diffusion coefficients D_ij

# for each element within each zone of the diffusion domain.#============================================================# Zone1

# Element1 xbegin = +0.0000e+000 xend = +3.2000e+002

Dmatrix(2, 2):

+4.8120e-014 -9.5250e-015 -2.8061e-015 +5.3131e-015 #============================================================# Zone2

# Element1 xbegin = +3.2000e+002 xend = +6.0000e+002

Dmatrix(2, 2):

+6.6662e-015 +1.2323e-015 -8.2553e-015 +1.7341e-014

Interpolation of Data

C C C

C’ C’ C’ C’

C1

1

2

2

3

3 4

4

Hermite Interpolation

In the experimental data, Ci are given. We have C´i = 0 at end points.

Fit Hermite interpolation polynomials over intervals.

This guarantees continuity of Ci and also of C´i across interpolation elements.

Multiphase Diffusion and Interpolation

Create several diffusion zones and interpolate separately.

Method of Finite ElementsSet up an action integral* for the differential equation under study. Break up the physical region into finite elements. The “PHYSICS” is the same in the elements as in the global system.Express the physical “field” in terms of unknown nodal values and interpolation polynomials.

Integrate out the spatial dependence and use a (“nodal”) variational method (Principle of Least Action) for determining the nodal variables.Derive the system of linear equations representing the original differential equation, and apply boundary conditions.Solve the linear equations and obtain nodal values. Reconstruct the full solution using the same interpolation functions as before.

*Use Galerkin’s method if no action principle exists.

Finite Element Representation of a Function

Element

y(x)

Nodes

y1 y

2

y(x) ~ y N (x) + y N (x)1 1 2 2~

1 2( ) 1 ; ( )N x x N x x= − =

Example: Linear Elements

Diffusion Equation

{ }( ) 0; / .2

jiij k

dCdC d D C x td d d

λ λλ λ λ

⎛ ⎞+ = =⎜ ⎟

⎝ ⎠

For FEA using Galerkin’s method, project the equation along test functions or shape functions. Express Ci in terms of shape functions and unknown nodal values Ciα for Ci and integrate out the coordinate dependence.

This leads to simultaneous equations for the nodal values of Ciα

0.2 i ij j

dN dNdNd N C d D Cd d d

β βαα β βλλ λ

λ λ λ⎛ ⎞ ⎛ ⎞

+ =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

∫ ∫

Concentration Continuity using Finite Element Overlay

Galerkin’s method leads to discretized diffusion equations. Now apply BC and solve linear equations.

M C rαβ β α=

Matrix equations are solved using sparse matrix methods.

Implementing BCs : Applying Matrix “Benediction”

C0 a1 0 0 0 …

Add column times (-a)

M C

Computational Issues

We have three diffusion equations for ternary diffusion, only two being relevant if mass conservation holds.

We use Hermite interpolation polynomials with 2 or 3 nodes per element:

• 4 x 4 or 6 x 6 element matrices

• With 2 equations for the 2 independent concentrations, we have 8 x 8 element matrices.

• Each diffusion partition may need 3~10 elements. If there are 10 partitions in the diffusion region, we have 30~100 elements, and a global matrix of dimension: 30x2 +2 = 62 to ~202.

So the FEM computational aspects are “managable.”